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Phase reduction and synchronization of a network of coupled dynamical elements exhibiting collective oscillations

Hiroya Nakao, Sho Yasui, Masashi Ota, Kensuke Arai, Yoji Kawamura
2018 Chaos  
A general phase reduction method for a network of coupled dynamical elements exhibiting collective oscillations, which is applicable to arbitrary networks of heterogeneous dynamical elements, is developed  ...  A set of coupled adjoint equations for phase sensitivity functions, which characterize the phase response of the collective oscillation to small perturbations applied to individual elements, is derived  ...  The only assumption is that the whole network has a stable limit-cycle solution.  ... 
doi:10.1063/1.5009669 pmid:31906627 fatcat:xoxprnx5mfevjjeeanlwy5prku

Multiscale modeling, stochastic and asymptotic approaches for analyzing neural networks based on synaptic dynamics

D. Holcman, F. Abergel, M. Aiguier, D. Challet, P.-H. Cournède, G. Faÿ, P. Lafitte
2014 ESAIM Proceedings and Surveys  
We discuss here recent progress about modeling and analysis of small and large neuronal networks. We present neural network modeling based on synaptic dynamics.  ...  Finally, this report illustrates how mathematical methods in neuroscience allows a better understanding of neural network dynamics.  ...  The phase space shows a limit cycle C (dashed line) containing a stable focus P 2 , a saddle point P S and a stable attractor P 1 . The Up-state is the domain inside the limit cycle C.  ... 
doi:10.1051/proc/201447003 fatcat:iaiv5kc3crgnnlc5eakshqirsq

Examining phase response curve of nerve cell by using three different methods

Hasan Eskalen, Şükrü Özğan
2018 International Journal of Chemistry and Technology  
Phase Response Curves (PRCs); act like a bridge between, a single neuron and neural network; briefly measure change in period of oscillation by giving perturbation at different points of oscillation.  ...  Neural system, that consists of billions of neurons are also exhibited periodic motion.  ...  Self-sustained oscillations can be represented geometrically by a stable limit cycle. 16 The stable limit cycles are stable against to small amplitude external forces.  ... 
doi:10.32571/ijct.338403 fatcat:4joiryb5lzfq3n2xy4es6jjrdm

Computational promise of simultaneous recurrent network with a stochastic search mechanism

2004 2004 IEEE International Joint Conference on Neural Networks (IEEE Cat No 04CH37541) IJCNN-04  
This paper explores the computational promise of enhancing Simultaneous Recurrent Neural networks with a stochastic search mechanism as static optimizers.  ...  Successful application of Simultaneous Recurrent Neural networks to static optimization problems, where the training had been achieved through one of a number of deterministic gradient descent algorithms  ...  Opinions, views and conclusions expressed are authors' only and do not reflect those of the NSF.  ... 
doi:10.1109/ijcnn.2004.1380969 fatcat:5cvava5w6rddddjodp54zrbimu


2001 International Journal of Neural Systems  
Recurrent backpropagation algorithm is employed to train the recurrent, relaxation-based neural network in order to associate fixed points of the network dynamics with locally optimal solutions of the  ...  A trainable recurrent neural network, Simultaneous Recurrent Neural network, is proposed to address the scaling problem faced by neural network algorithms in static optimization.  ...  of stable equilibrium points, limitations and bounds on weight update formulae.  ... 
doi:10.1142/s012906570100062x pmid:11709814 fatcat:5663wp4p6zgdtbavvfxag6bvqe

Macroscopic phase resetting-curves determine oscillatory coherence and signal transfer in inter-coupled neural circuits [article]

Gregory Dumont, Boris Gutkin
2018 arXiv   pre-print
Various cellular level mechanisms influence the network dynamics and structure the macroscopic firing patterns.  ...  From there we determine the structure of macroscopic coherence states (phase locking) of two weakly synaptically-coupled networks.  ...  from the limit cycle.  ... 
arXiv:1812.03455v1 fatcat:2oc4v7hf2raupf2fdqp5uffbxu

Macroscopic phase resetting-curves determine oscillatory coherence and signal transfer in inter-coupled neural circuits

Grégory Dumont, Boris Gutkin, Daniele Marinazzo
2019 PLoS Computational Biology  
While multiple cellular mechanisms influence the network oscillatory dynamics and structure the macroscopic firing motifs, one of the key questions is to identify the biophysical neuronal and synaptic  ...  A second important issue is how the different neural activity coherence states determine the communication between the neural circuits.  ...  Assuming that the reduced E-I system (2) and (3) has a stable limit cycle, we find that (see Methods for more detail) the iPRC Z(t) is a periodic vector that is a solution of the adjoint equation À d dt  ... 
doi:10.1371/journal.pcbi.1007019 pmid:31071085 pmcid:PMC6529019 fatcat:moxsgxm2jrazxatuloggym5654

Synchronization of electrically coupled resonate-and-fire neurons [article]

Thomas Chartrand, Mark S. Goldman, Timothy J. Lewis
2018 arXiv   pre-print
We calculate the phase response curve using an extension of the adjoint method that accounts for the discontinuity in the dynamics.  ...  Using the theory of weakly coupled oscillators, we explore the effect of both subthreshold and spike-mediated coupling on synchrony in small networks of electrically coupled resonate-and-fire neurons,  ...  We proceed by first evaluating the adjoint equation (3.8) for the subthreshold dynamics. In general, the adjoint equation evaluates the dynamics linearized about the limit cycle.  ... 
arXiv:1801.05874v1 fatcat:vl3b43mqmnai3i2pga5v6yljnq

The Effects of Spike Frequency Adaptation and Negative Feedback on the Synchronization of Neural Oscillators

Bard Ermentrout, Matthew Pascal, Boris Gutkin
2001 Neural Computation  
This latter mechanism applies with a network that has recurrent inhibition. The shunting behavior is captured in a simple two-variable reduced model that arises near certain types of bifurcations.  ...  We show that both of these have strong effects on the synchronization properties of excitatorily coupled neurons. Furthermore, we show that the reasons for these effects are different.  ...  Acknowledgments The research reported in this article was supported by the National Institute of Mental Health and the National Science Foundation.  ... 
doi:10.1162/08997660152002861 pmid:11387047 fatcat:ga46d2gv4vg5fdkbpoawzke6om

The role of node dynamics in shaping emergent functional connectivity patterns in the brain [article]

Michael Forrester, Stephen Coombes, Jonathan J. Crofts, Stamatios N. Sotiropoulos, Reuben D. O'Dea
2019 arXiv   pre-print
dynamics of a single node, highlighting a non-trivial structure--function relationship in regimes that support limit cycle oscillations.  ...  We treat a system of Jansen--Rit neural-mass nodes with heterogeneous structural connections estimated from diffusion MRI data provided by the Human Connectome Project.  ...  The first is the so-called phase response or adjoint Q ∈ R m , that describes the response of an attracting limit cycle to a small perturbation. This can be computed by solving the adjoint equation.  ... 
arXiv:1906.11573v1 fatcat:oobvnqxyjnfdvn72wki4xfiqya

Learning Dynamics Models with Stable Invariant Sets [article]

Naoya Takeishi, Yoshinobu Kawahara
2020 arXiv   pre-print
In this paper, we propose a method to ensure that a dynamics model has a stable invariant set of general classes such as limit cycles and line attractors.  ...  Invariance and stability are essential notions in dynamical systems study, and thus it is of great interest to learn a dynamics model with a stable invariant set.  ...  stable invariant sets (e.g., limit cycles, limit tori, and continuous sets of infinitely many equilibria).  ... 
arXiv:2006.08935v2 fatcat:ovchisezfbbqloa74rwqy3z6ra

Response of traveling waves to transient inputs in neural fields

Zachary P. Kilpatrick, Bard Ermentrout
2012 Physical Review E  
This wave response function is analogous to the phase response curve of limit cycle oscillators.  ...  Neural fields are modeled as integrodifferential equations whose convolution term represents the synaptic connections of a spatially extended neuronal network.  ...  There is a clear analogy between the response function we have derived for waves and the phase response curve of limit cycle oscillators [53] .  ... 
doi:10.1103/physreve.85.021910 pmid:22463247 fatcat:ruakkfmfrvepncdbuj4vuefxe4

Learning Stable Galerkin Models of Turbulence with Differentiable Programming [article]

Arvind T. Mohan, Kaushik Nagarajan, Daniel Livescu
2021 arXiv   pre-print
, when compared to purely data-driven neural networks.  ...  Despite the complexity of building data-driven ROMs for turbulence, the superior representational capacity of deep neural networks has demonstrated considerable success in learning ROMs.  ...  between neural networks and ODEs.  ... 
arXiv:2107.07559v1 fatcat:z7erb76iinbtflmmkneckdrmj4

Dimension Reduction of Biological Neuron Models by Artificial Neural Networks

Kenji Doya, Allen I. Selverston
1994 Neural Computation  
This gives rise to a stable limit cycle that goes back and forth between the two sections of the fast nullclines.  ...  Then the trajectory heads to- ward the stable limit cycle around a, which is located on the left-hand side of the slow nullcline where y3; > 0.  ... 
doi:10.1162/neco.1994.6.4.696 fatcat:gfalfsx2onhxxbqegxdf7nradq

Layer-Parallel Training of Deep Residual Neural Networks [article]

S. Günther, L. Ruthotto, J.B. Schroder, E.C. Cyr, N.R. Gauger
2019 arXiv   pre-print
Residual neural networks (ResNets) are a promising class of deep neural networks that have shown excellent performance for a number of learning tasks, e.g., image classification and recognition.  ...  Hence, training a ResNet can be cast as an optimal control problem of the associated dynamical system.  ...  This motivates limiting the forward propagation to stable dynamics, e.g., inspired by hyperbolic systems.  ... 
arXiv:1812.04352v3 fatcat:lwtg3ko7xvae3dpilndk6i5voq
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